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Complex absorbing potential method for Stark resonances

Kentaro Kameoka

Abstract

We characterize the resonances of Stark Hamiltonians by the complex absorbing potential method. Namely, we prove that the Stark resonances are the limit points of complex eigenvalues of the Stark Hamiltonian with a quadratic complex absorbing potential when the absorbing coefficient tends to zero. The proof employs the complex distortion outside a cone introduced in the previous work by the author. Potentials with local singularities such as the Coulomb potential are allowed as perturbations.

Complex absorbing potential method for Stark resonances

Abstract

We characterize the resonances of Stark Hamiltonians by the complex absorbing potential method. Namely, we prove that the Stark resonances are the limit points of complex eigenvalues of the Stark Hamiltonian with a quadratic complex absorbing potential when the absorbing coefficient tends to zero. The proof employs the complex distortion outside a cone introduced in the previous work by the author. Potentials with local singularities such as the Coulomb potential are allowed as perturbations.
Paper Structure (9 sections, 7 theorems, 54 equations)

This paper contains 9 sections, 7 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Complex distorted operator and complex absorbing potential
  3. Justification of complex absorbing potential method for Stark Hamiltonians
  4. Resolvent estimate for the free distorted operator with complex absorbing potential
  5. Proof of convergence to Stark resonances
  6. Proofs of technical lemmas
  7. Proof of Lemma \ref{['estimate-1']}
  8. Proof of Lemma \ref{['estimate-2']}
  9. Local singularities

Key Result

Theorem 1

Suppose Assumption 1 holds. Then for any $\chi_1, \chi_2 \in L_{\mathrm{comp}}^{\infty}(\mathbb{R}^n)$ such that $\chi_j=1$ near $\mathrm{supp}\,V_{\mathrm{sing}}$, the cutoff resolvent $\chi_1 R_{+}(z) \chi_2 \mspace{7mu}, (\mathrm{Im}z>0)$, has a meromorphic continuation to $\{z|\,\mathrm{Im}z>-\d The set of resonances is independent of the choices of $\chi_1$, $\chi_2$ including multiplicities

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Example 1.1
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof : Proof of Theorem \ref{['thm-1']}
  • ...and 6 more